17. Suppose that an experiment can result in one of r possible outcomes, the ith outcome having probability pi , i = 1,...,r, r i=1 pi = 1. If n of these experiments are performed, and if the outcome of any one of the n does not affect the outcome of the other n − 1 experiments, then show that the probability that the first outcome appears x1 times, the second x2 times, and the rth xr times is n!

x1! x2! ... xr!

px1 1 px2 2 ··· pxr r when x1 + x2 +···+ xr = n This is known as the multinomial distribution.